On impulsive optimal control with functions of not bounded variation

Question

I have the following optimal control problem $$ J=\int_0^TF(t,y_1(t),y_2(t))dt \to \min, $$ subject to \begin{align} &\dot y_1(t) = f(t,y_1(t),y_2(t)) + g(t)\nu(t),\\ &\dot y_2(t) = \nu(t), \end{align} where $\nu$ is an impulsive control, and $y_i$ are phase coordinates. Results on necessary conditions (maximum principle) of this type of problems which I've seen require functions $f$ and $g$ be smooth in $t$. But in my case both of them have unbounded variation in $t$ in any interval (trajectory of Brownian motion as a particular case).

So how this problem can be solved and are there any result on this type of problems?

Answer 1

In case someone else is interested, there is a paper by Arutyunov, Karamzin and Pereira, "Pontryagin’s maximum principle for constrained impulsive control problems", which presents a version of maximum principle for problems with non-smooth data. Here is a link for it on ScineceDirect.